Question

1. find the center of the circle with endpoints (6, - 8) and (4, - 4)

a. (2, - 4)
b. (10, - 12)
c. (5, - 6)
d. (5, - 4)
e. (10, - 6)

1. determine whether this relation is a function. {(2, 10), (3, - 8), (4, 6), (5, 10)}

a. yes, it is a function.
b. no, it is not a function.

1. let f(x)=6x² + 3. find the difference quotient (f(x + h)-f(x))/h, h≠0.

a. 3x
b. 12x + 6h
c. 12x
d. 6x + 6h
e. 6x

1. find the equation of the line in slope - intercept form that has a slope of - 5 and passes through the point (- 4, - 2).

a. y=-5x - 10
b. y=-5x + 2
c. y=-5x + 4
d. y=-5x - 22
e. y=-4x - 2

1. find the average rate of change of f(x)=x² + 4 on 1, 4.

a. 15
b. 10
c. 5
d. 0
e. - 5

Explanation:

Step1: Recall mid - point formula

The center of a circle with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is the mid - point \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Given \((x_1,y_1)=(6,-8)\) and \((x_2,y_2)=(4,-4)\), then \(x=\frac{6 + 4}{2}\) and \(y=\frac{-8+( - 4)}{2}\).

Step2: Calculate \(x\) - coordinate of the center

\(x=\frac{6 + 4}{2}=\frac{10}{2}=5\).

Step3: Calculate \(y\) - coordinate of the center

\(y=\frac{-8-4}{2}=\frac{-12}{2}=-6\). So the center is \((5,-6)\).

Step4: Check function definition

A relation \(\{(x_1,y_1),(x_2,y_2),\cdots\}\) is a function if for each \(x\) - value there is exactly one \(y\) - value. In the relation \(\{(2,10),(3,-8),(4,6),(5,10)\}\), each \(x\) value \(2,3,4,5\) has a unique \(y\) - value associated with it, so it is a function.

Step5: Find difference quotient

Given \(f(x)=6x^{2}+3\), then \(f(x + h)=6(x + h)^{2}+3=6(x^{2}+2xh+h^{2})+3=6x^{2}+12xh + 6h^{2}+3\).
\(\frac{f(x + h)-f(x)}{h}=\frac{(6x^{2}+12xh + 6h^{2}+3)-(6x^{2}+3)}{h}=\frac{6x^{2}+12xh + 6h^{2}+3 - 6x^{2}-3}{h}=\frac{12xh+6h^{2}}{h}=12x + 6h\).

Step6: Find line equation

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. Given \(m=-5\) and the point \((x,y)=(-4,-2)\), substitute into \(y=mx + b\): \(-2=-5\times(-4)+b\), \(-2 = 20 + b\), then \(b=-2 - 20=-22\). So the equation is \(y=-5x-22\).

Step7: Find average rate of change

The average rate of change of a function \(y = f(x)\) on the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Given \(f(x)=x^{2}+4\), \(a = 1\), \(b = 4\), \(f(1)=1^{2}+4=5\), \(f(4)=4^{2}+4=16 + 4=20\). Then \(\frac{f(4)-f(1)}{4 - 1}=\frac{20 - 5}{3}=\frac{15}{3}=5\).

Answer:

1. C. \((5,-6)\)
2. A. Yes, it is a function.
3. B. \(12x + 6h\)
4. D. \(y=-5x-22\)
5. C. \(5\)